A cone made of insulating material has a total charge Q spread uniformly over its sloping surface. Calculate the work done in bringing a small test charge q from infinity to the apex of the cone. The cone has a slope length L.

Explanation

`W_("ext") = Delta U = U_f - U_ i = q (V_A - V_oo)`
`W_("ext") = q (V_A - 0) = qV_A`...(i)
Here `V_A` is electric potential at point A. The electric charge on the considered ring is
`d q = (Q 2 pi r d x)/(pi R L)`
.
Therefore, electric potential due to considered ring is
`d V = (d q)/(4 pi epsilon_0 x)`
`d V = (Q 2 pi r d x)/(pi R L 4 pi epsilon_0 x)`
`:' sin theta = r/(x) = (R)/(L) :. r = (R x)/(L)`
`:. d V = (2 pi Q(R/L x)d x)/(pi R L 4 pi epsilon_0 x)`
`V_A = int d v = q/(2 pi epsilon_0 L^2) underset(0)overset(L)(int) dx = Q/(2 pi epsilon_0) L`
`:. W_("ect") = q V_A = (q Q)/(2 pi epsilon_0 L)` from eqn...(i).